Description
Functions whose Fourier transforms have compact supports, called band-limited, are widely used in various engineering fields, especially in signal processing. According to the well-known Nyquist-Shannon sampling theorem, if we have a deterministic (discrete) equidistant sample from this function having a sampling rate at least twice the bandwidth (Nyquist rate), then the function can be perfectly reconstructed from the sample. It this talk we investigate the cases, when we only have a finite dataset randomly sampled from a (potentially unknown) input distribution, and the outputs of the target (band-limited) function can also have measurement noises. We suggest algorithms for these four cases, i.e., known / unknown input distribution, and function values with / without measurement noises, which are able to construct simultaneous confidence bands for the underlying true (regression) function under very mild statistical assumptions; they are essentially distribution-free. We present non-asymptotic theoretical guarantees for the cases of known input distributions (unknown measurement noises can still affect the outputs). For the case of unknown input distributions, the confidence bands are asymptotically guaranteed. We argue that all of these (four) methods are uniformly strongly consistent, i.e., their confidence bands shrink around the true regression function and in the limit they cannot contain any other functions, almost surely. Finally, we illustrate the ideas via a series of numerical experiments, including applications to image restoration and super-resolution.
Joint work with Bálint Horváth (BME).